{"id":"fbbdaae6-c0b4-4400-8738-30b854f582e9","arxiv_id":"2502.05132","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Number-fluctuation thermometry based on the exact density correlation function extracts global and local temperatures of a quasi-2D ideal Fermi gas from single-atom-resolved images, without invoking the fluctuation-dissipation theorem.","lead":"The authors measure atom number fluctuations in tiny probe volumes of an ultracold Fermi gas imaged atom by atom, and use the exact relation between fluctuations and density correlations to extract temperature without relying on the usual fluctuation-dissipation theorem. The method gives global and local temperatures over a wide range and reveals sub-extensive fluctuation corrections at low temperature.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The inversion to T is only valid if each probe volume is locally in thermal equilibrium; this condition is acknowledged after Eq. (2) but the paper's 'far out-of-equilibrium' and 'nearly universal' claims go beyond the demonstrated evidence.","rationale":"The central claim is conditional: Eq. (2) is exact, but using it as a thermometer requires a known g2(T,n). For the ideal Fermi gas, g2 is known and the experiment validates the variance-curve collapse over T/TF = 0.29(1) to 3.6(3). The g2 comparison in Fig. S2 is a strong, non-circular check: the measured pair correlations agree with theory at the temperatures extracted from fluctuations, which directly supports local thermal equilibrium in the central region. I therefore do not view the local-equilibrium assumption as a fatal flaw for the demonstrated system; it is the boundary of the paper's broader claims. The abstract and conclusion reach beyond this boundary by suggesting applicability to arbitrary far-from-equilibrium gases, for which there is no evidence that g2 remains a function of T only. The reader's weakest assumption identifies the same issue. The conditional verdict is appropriate: with a strengthened local-equilibrium test across the full cloud and a more carefully scoped generality claim, the paper would be a clear accept.","tokens_in":15014,"tokens_out":9979,"duration_ms":110016,"concrete_test":"Reanalyze the existing images by dividing each cloud into several radial annuli. In each annulus, fit T from the local ΔN²(⟨N⟩) curve using Eq. (2), then compute the theoretical g2(r) at that T and compare it with the directly measured g2(r) in the same annulus. Local equilibrium is confirmed only if the same T simultaneously reproduces both observables in every annulus; any annulus where the best-fit T fails to predict its measured g2 would indicate a violation of the local-equilibrium assumption. A complementary test would be to apply the same analysis to a deliberately non-thermalized cloud (e.g., t_therm = 0.2 s after modulation) and check whether the ΔN²-versus-⟨N⟩ points collapse onto a single-temperature curve.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (2) maps (⟨N⟩, ΔN²) to T only when the reduced correlation function g2(r1,r2) is a function of local density and temperature alone. For an ideal Fermi gas in the grand-canonical ensemble this holds, and the data in Fig. 2 and Fig. S2 support it. The load-bearing condition is the local-equilibrium assumption stated after Eq. (2) ('provided it is locally at thermal equilibrium'). The experimental support for this condition is incomplete relative to the breadth of the claims: the thermalization measurement in Fig. S1 is performed only for the longest modulation (160 ms) and monitors only the in-plane width σxy; the collapse of all variance data onto a single curve is an indirect, necessary test; the g2 comparison in the first row of Fig. S2 is shown only in the central region of the cloud. If a subsystem has a non-thermal momentum distribution, g2 is not a local function of T, and Eq. (2) cannot be inverted to a thermodynamic temperature. The manuscript's abstract claims the method 'does not require global thermal equilibrium' and is a 'nearly universal thermometer,' and the conclusion says it extends thermometry to gases 'far out-of-equilibrium.' These statements go beyond the presented demonstration, which establishes the method for locally equilibrated ideal Fermi gases. This is a scope limitation rather than an internal inconsistency: the central derivation and the ideal-gas demonstration are sound.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper introduces a thermometry method for ultracold quantum gases based on the exact relation between atom-number fluctuations in a probe volume and the two-particle density correlation function g2 (Eq. (2)). The authors demonstrate the method on a quasi-two-dimensional ideal Fermi gas imaged with continuum quantum gas microscopy. They extract global temperatures from fits of the variance-versus-mean curve with temperature as the sole free parameter, over reduced temperatures T/T_F from 0.29(1) to 3.6(3), and verify the results by comparing the measured g2 to theory without additional fitting. They further demonstrate local thermometry down to probe volumes comparable to the Fermi-hole diameter, and use the method to isolate a sub-extensive contribution ΔQ to the number fluctuations beyond the fluctuation-dissipation term, comparing it to numerical and asymptotic predictions.","tokens_in":15232,"tokens_out":4685,"duration_ms":48887,"significance":"If the central claims hold, the paper provides a valuable addition to ultracold-atom thermometry: it replaces the fluctuation-dissipation approximation with an exact correlation-based identity, works for arbitrary trapping geometries without precise trap calibration, and gives local temperatures at the scale of the correlation length. The experimental validation is notably strong in several respects: the Eq. (2) identity is exact and cleanly derived; the classical-gas data fall on the parameter-free Poissonian line; the g2 comparison in Fig. S2 is made without fitting parameters; and the dynamic range of degeneracy is broad. These strengths make the core method credible for locally equilibrated ideal Fermi gases.","major_comments":[{"comment":"The claims that the method does not require global thermal equilibrium and extends thermometry to gases 'far out-of-equilibrium' overstate what is demonstrated. The inversion of Eq. (2) to a temperature is valid only 'provided it is locally at thermal equilibrium' (stated immediately after Eq. (2)), and this condition is checked in Fig. S1 only for the longest modulation time (160 ms) and only through the in-plane width σxy. The collapse of the variance data onto a single curve is an indirect necessary condition, not a sufficient test of local equilibrium; a non-thermal momentum distribution would make g2 depend on more than (n, T) and render the extracted 'temperature' ambiguous. The authors should qualify the universality claims to locally equilibrated systems or provide additional evidence supporting the broader 'far out-of-equilibrium' statement.","section":"Abstract, Discussion, Conclusion; after Eq. (2)"},{"comment":"The agreement between the measured ΔQ and the theoretical prediction from Eq. (5) is partly by construction. The measured ΔQ is obtained by subtracting kBT ∂⟨N⟩/∂μ|T from the measured ΔN², using the temperature T extracted from the variance fit to Eq. (2). Because Eq. (4) is an exact identity, this construction forces the measured ΔQ to match the right-hand side of Eq. (5) evaluated at the same T and density. The 'without fitting parameter' statement in the Fig. 4 caption is technically true but the comparison is a consistency check, not an independent validation of the theory. The empirical observation that ΔQ is positive and decreases with T remains valid, but the claim of 'excellent agreement' should be reframed, or an independently determined T (for example from the g2 comparison in Fig. S2) should be used to construct both the data points and the theory curve.","section":"Fig. 4 and Eq. (5)"}],"minor_comments":[{"comment":"Typo: 'reprents' should be 'represents'.","section":"Fig. 3 caption"},{"comment":"Typo: 'spacial' should be 'spatial'.","section":"Supplementary Materials, 'Influence of the Probe Volume'"},{"comment":"The assertion that 'for a given ⟨N⟩, ΔN² monotonically increases with T' is used to justify the uniqueness of the inversion, but no proof or reference is provided; a brief derivation or citation would make the uniqueness claim rigorous.","section":"Fig. 1b caption"},{"comment":"The statement that 'there is no formal limitation to how small the system S can be' is later qualified in the Supplementary Materials by the pinning-lattice constraint (L ≳ aL); the wording in the main text should be reconciled with this practical limitation.","section":"Local Thermometry section"},{"comment":"The g2 comparison in the first row of Fig. S2 is shown only for the central region of the cloud; a sentence stating whether the agreement holds across the full cloud would clarify the spatial robustness of the temperature determination.","section":"Fig. S2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a good experimental methods paper. The core idea—use the exact relation between number variance and g2 to extract T from atom-resolved images—is not new in itself (Castin's notes have it), but the demonstration as a working thermometer in a continuum gas, with local temperatures down to the Fermi-hole scale, is new and convincing. The experimental checks are the best part: the classical gas sits on the Poissonian line, the measured g2 matches the theory at the fitted T with no free parameters, and the T/T_F range 0.29 to 3.6 is genuinely broad.\\nThe weak spot is the framing. Eq. (2) is only a thermometer if g2 is a known function of density and temperature, which in practice means the probe volume is locally in thermal equilibrium. The authors acknowledge this after Eq. (2) and check thermalization for the longest modulation time, but only through the in-plane width relaxation. That is indirect evidence. The abstract's 'does not require global thermal equilibrium' and the conclusion's 'far out-of-equilibrium' claims go beyond what is shown. Nothing in the data demonstrates the method on a system without local equilibrium, and for a non-thermal momentum distribution g2 would not have the simple (n,T) dependence required for the inversion. This is a scope limitation, not an internal inconsistency.\\nThe ΔQ measurement in Fig. 4 is a bit circular for my taste: the same fitted T goes into computing the extensive term that is subtracted off, and then the theory curve for ΔQ is evaluated at that same T. It's a consistency check, not an independent confirmation. Still, the data collapse and the no-fit theory comparison are convincing that the sub-extensive contribution is real.\\nWho is this for? Groups working with quantum gas microscopes and anyone measuring temperatures in cold atoms. The method itself is limited to systems where g2 is computable, but that includes a wide class of interesting models. The paper deserves a serious referee. It will need revision to align the claims with the evidence, but the core result is solid.","headline":"A convincing demonstration of correlation-based fluctuation thermometry on an ideal Fermi gas, with the main caveat that the advertised universality and out-of-equilibrium reach outpace the evidence.","tokens_in":15846,"tokens_out":3996,"would_cite":true,"duration_ms":34530,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Atom-count noise alone fixes the temperature of a quantum gas.","keywords":["ultracold Fermi gas","quantum gas microscopy","fluctuation thermometry","density-density correlations","sub-extensive fluctuations","local thermometry","grand-canonical ensemble","g2 correlation function"],"falsifier":"Prepare the same cloud, modulate the light sheet for 160 ms, but hold for only a short time (well under the measured 1.4 s relaxation) before removing one spin and imaging; under local equilibrium the temperatures extracted from probe volumes at different positions should disagree and track the vertical-energy imbalance, so any single consistent temperature across all probes would contradict the assumption and invalidate the inversion.","tokens_in":14769,"feed_emoji":"🌡️","tokens_out":13225,"duration_ms":114058,"temperature":0.7,"pith_summary":"This paper introduces a thermometer for ultracold quantum gases that reads the temperature directly from single-atom-resolved images, without fitting the trapping potential and without assuming the whole cloud is one thermal sample. The central quantity is the exact relation between the atom-number variance $\\Delta N^2$ in a small probe volume and the two-atom correlation function $g_2$; when $g_2$ is known as a function of density and temperature, a measured pair $(\\langle N\\rangle, \\Delta N^2)$ corresponds to exactly one temperature. On a quasi-two-dimensional ideal Fermi gas, the authors demonstrate this over reduced temperatures $T/T_F$ from $0.29(1)$ to $3.6(3)$, and show that the same variance curve gives consistent global and local temperatures down to the scale of the inter-particle spacing. The method also separates the fluctuation-dissipation contribution from a sub-extensive part $\\Delta Q$, which is shown to match an exact cross-correlation formula without fitting. If these results hold, thermometry of quantum gases becomes local, calibration-free, and applicable to homogeneous or out-of-equilibrium systems.","feed_headline":"Atom-count noise alone fixes the temperature of a quantum gas","feed_subtitle":"One variance-vs-count curve sets local and global T from 0.29 to 3.6 T/TF, with no trap fit.","key_machinery":"The load-bearing identity is Eq. (2): the variance of the atom number in $S$ is rewritten exactly as $\\langle N\\rangle$ minus the integrated pair-exclusion integral $n^2\\int_S\\int_S[1-g_2(r_1,r_2)]\\,dr_1dr_2$, where $g_2$ is the normalized two-point density-density correlation function. Because $g_2$ for the ideal Fermi gas is known and depends on the dimensionless products $k_F r$ and $T/T_F$, the variance becomes a function of $\\langle N\\rangle$ and $T$ only, and the monotone rise of $\\Delta N^2$ with $T$ at fixed $\\langle N\\rangle$ makes the inversion unique. The local-density approximation then reads the trapped cloud as many homogeneous samples, so a single modulation-time preparation yields a full $\\Delta N^2$-versus-$\\langle N\\rangle$ curve, and the vertical motion is handled by summing the variances of the occupied $z$-levels with populations $p_\\nu$ fixed by $\\mu$, $T$, and $\\omega_z$. For the sub-extensive part, the same correlation function is integrated across the boundary between $S$ and its complement, giving Eq. (5) for $\\Delta Q$.","core_discovery":"On its own terms, the paper's discovery is that the exact identity $$\\$\\Delta$ $N^{2}$ = \\langle N\\rangle - $n^{2}$ \\int_S \\int_S [1-g_2(r_1,r_2)]\\,dr_1 dr_2$$ turns two measured numbers, the average atom number $\\langle N\\rangle$ and the variance $\\Delta N^2$ in a probe volume $S$, into a one-to-one thermometer. For an ideal Fermi gas, $g_2$ depends only on $k_F r$ and $T/T_F$, hence only on $\\langle N\\rangle$ and $T$, so each measured pair selects a unique temperature. The authors verify their temperatures by computing $g_2$ from the same images at the extracted $T$ and finding agreement without any fitting parameter, and they apply the method at probe sizes down to the Fermi-hole diameter. Separately, they isolate the sub-extensive correction $\\Delta Q = \\Delta N^2 - k_B T\\, \\partial\\langle N\\rangle/\\partial\\mu|_T$, derive the exact expression $\\Delta Q = n^2 \\int_S dr_1 \\int_{\\overline S} dr_2\\,[1-g_2(r_1,r_2)]$, and show that the measured $\\Delta Q$ follows that expression from the quantum-degenerate to the classical regime. The paper's broader assertion is that any system whose density-density correlation function can be computed, analytically or numerically, inherits a local and global thermometer from this relation alone.","pith_inferences":["A natural extension the authors leave implicit: at known temperature, the variance curve can be inverted to constrain $g_2$ itself, turning the apparatus into a correlation-function microscope for interacting systems where $g_2$ is not known analytically.","The uniqueness of the inversion relies on monotonicity of $\\Delta N^2$ with $T$ at fixed $\\langle N\\rangle$; systems with non-monotonic variance, for instance near a phase transition, would need a third joint observable to select $T$.","Because $\\Delta Q$ is a cross-correlation between the probe and its complement, spatially resolved measurements of it could map non-local or entanglement-related fluctuations across a sample, connecting this thermometer to quantum-information observables.","A direct testable extension is to apply the method while the cloud is still relaxing after the heating modulation and check whether the local-temperature map tracks the vertical-energy imbalance; disagreement would quantify how far the local-equilibrium assumption is from holding."],"forward_implications":["The global temperature of a trapped cloud is obtained from one fit of all probe volumes simultaneously, making the result stable against local deviations and sensitive to incomplete thermalization.","Local thermometry at the scale of the Fermi hole is demonstrated for $T/T_F$ from $0.29(1)$ to $2.6(2)$, so temperature maps can be built across a spatially inhomogeneous sample.","Because the method requires neither global equilibrium nor trap calibration, it extends to homogeneous systems and to out-of-equilibrium configurations such as quenches or heat-transport experiments, where standard density-profile fitting fails.","The measured $\\Delta Q$ and the exact formula (5) give an experimental window on sub-extensive fluctuations, showing that the fluctuation-dissipation term alone is insufficient in small probe volumes at low temperature.","Any system with a computable $g_2$, including interacting one-dimensional gases, two-dimensional BKT superfluids, and lattice or continuum models accessible to quantum Monte Carlo, inherits this thermometer, as do phase-separated systems whose outer trivial reservoir can serve as the probe."],"supporting_citations":[{"why":"Supplies the known $g_2$ and thermodynamic relations for the ideal Fermi gas that enter the central Eq. (2).","marker":"[28]"},{"why":"Introduces the single-atom imaging and pinning protocol that provides the resolved number counts used throughout.","marker":"[29]"},{"why":"Provides the continuum quantum-gas-microscopy imaging and the $g_2$ formalism for quasi-2D fermions used to close the thermometer.","marker":"[30]"},{"why":"Defines the fluctuation-dissipation thermometry that the new method replaces; its limitations motivate the exact correlation route.","marker":"[2]"},{"why":"Gives the asymptotic $L^{d-1}\\log L$ scaling of $\\Delta Q$ at zero temperature and the thermodynamic fluctuation framework.","marker":"[32]"},{"why":"Provides the finite-temperature analysis of number fluctuations and compressibility used to interpret the $\\Delta Q$ behavior.","marker":"[33]"},{"why":"Defines the quantum variance that $\\Delta Q$ coincides with at $T=0$, giving the sub-extensive fluctuations their physical interpretation.","marker":"[36]"},{"why":"Demonstrates atom-resolved fluctuation thermometry on a lattice using the generalized fluctuation-dissipation theorem, the closest predecessor at single-atom resolution.","marker":"[21]"},{"why":"Shows the analogue non-local fluctuation effects in Fermi-Hubbard gases that motivate the sub-extensive measurement.","marker":"[19]"}],"fun_headline_variants":["Number variance alone sets quantum gas temperature","Exact relation makes atom counts a local and global thermometer","Thermometer for quantum gases skips trap calibration","Beyond fluctuation-dissipation: exact number-fluctuation thermometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The method assumes each atom-resolved probe volume is locally at thermal equilibrium at one temperature, so the two-atom correlation function $g_2$ depends only on density and that temperature; if local equilibrium fails, the measured pair $(\\langle N\\rangle, \\Delta N^2)$ cannot be inverted to a unique $T$.","fun_headline_variants_meta":{"raw":{"variants":["Number variance alone sets quantum gas temperature","Exact relation makes atom counts a local and global thermometer","Thermometer for quantum gases skips trap calibration","Beyond fluctuation-dissipation: exact number-fluctuation thermometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00024,"raw_usage":{"total_tokens":1592,"prompt_tokens":1091,"completion_tokens":501,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":707,"completion_tokens_details":{"reasoning_tokens":446}},"tokens_in":707,"tokens_out":501,"duration_ms":5616,"temperature":1.0,"reasoning_tokens":446,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T20:07:35.477392+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Prepare the same cloud, modulate the light sheet for 160 ms, but hold for only a short time (well under the measured 1.4 s relaxation) before removing one spin and imaging; under local equilibrium the temperatures extracted from probe volumes at different positions should disagree and track the vertical-energy imbalance, so any single consistent temperature across all probes would contradict the assumption and invalidate the inversion.","supporting_citations":[{"cited_title":"Castin, in Lecture Notes of the 2006 Varenna Enrico Fermi School on Fermi Gases , edited by M","cited_arxiv_id":null,"evidence_quote":"Supplies the known $g_2$ and thermodynamic relations for the ideal Fermi gas that enter the central Eq. (2)."},{"cited_title":"Verstraten, K","cited_arxiv_id":null,"evidence_quote":"Introduces the single-atom imaging and pinning protocol that provides the resolved number counts used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the asymptotic $L^{d-1}\\log L$ scaling of $\\Delta Q$ at zero temperature and the thermodynamic fluctuation framework."},{"cited_title":"Klawunn, A","cited_arxiv_id":null,"evidence_quote":"Provides the finite-temperature analysis of number fluctuations and compressibility used to interpret the $\\Delta Q$ behavior."},{"cited_title":"Fr´ erot and T","cited_arxiv_id":null,"evidence_quote":"Defines the quantum variance that $\\Delta Q$ coincides with at $T=0$, giving the sub-extensive fluctuations their physical interpretation."},{"cited_title":"Hartke, B","cited_arxiv_id":null,"evidence_quote":"Demonstrates atom-resolved fluctuation thermometry on a lattice using the generalized fluctuation-dissipation theorem, the closest predecessor at single-atom resolution."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows the analogue non-local fluctuation effects in Fermi-Hubbard gases that motivate the sub-extensive measurement."}],"review_version":1}